Density of classical points in eigenvarieties
Loeffler, David. (2011) Density of classical points in eigenvarieties. Mathematical Research Letters, Vol.18 (No.5). pp. 983-990. ISSN 1945-001X
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In this short note, we study the geometry of the eigenvariety parametrizing p-adic automorphic forms for GL(1) over a number field, as constructed by Buzzard. We show that if K is not totally real and contains no CM subfield, points in this space arising from classical automorphic forms (i.e. algebraic Grossencharacters of K) are not Zariski-dense in the eigenvariety (as a rigid space); but the eigenvariety posesses a natural formal scheme model, and the set of classical points is Zariski-dense in the formal scheme.
We also sketch the theory for GL(2) over an imaginary quadratic field, following Calegari and Mazur, emphasizing the strong formal similarity with the case of GL(1) over a general number field.
|Item Type:||Journal Article|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||Faculty of Science > Mathematics|
|Library of Congress Subject Headings (LCSH):||Automorphic forms|
|Journal or Publication Title:||Mathematical Research Letters|
|Official Date:||September 2011|
|Page Range:||pp. 983-990|
|Access rights to Published version:||Restricted or Subscription Access|
|Funder:||Engineering and Physical Sciences Research Council (EPSRC)|
|Grant number:||EP/F04304X/2 (EPSRC)|
 K. Buzzard, On p-adic families of automorphic forms, in Modular curves and abelian varieties
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